Three-wave resonance interaction of disturbances in a boundary layer

Abstract

The three-frequency resonance of Tolman-Schlichting waves, one of which propagates along the stream while the other two propagate at adjacent angles to it, is investigated as a function of the spectrum and initial intensity in incompressible flows of the boundary-layer type within the framework of a weakly nonlinear theory. In the parallel-flow approximation such an interaction leads to the formation of unstable self-oscillations. The spatial evolution of the associated disturbances is studied with allowance for the self-similar deformation of the velocity profile of the main flow. It is shown that such development can lead to a sharp amplification of the oscillations, primarily of those propagating at an angle to the flow. The role of the effects under consideration in the transitional process and the connection with experimental data are discussed. As experiments [1, 2] show, in the process of a transition from a laminar boundary layer to a turbulent region, well described by the linear theory of hydrodynamic stability, there first comes a section of the excitation of harmonics of a Tolman-Schlichting wave, the appearance of three-dimensional structures, and a rapid growth in the intensity of low-frequency oscillations. There is no doubt that in this section the phenomena are dependent on the nonlinear character of the development with disturbances. The resonance interaction of wave triads can play an important role in this. For small enough amplitudes such an interaction is described by a first-order theory [3, 4], and in the general case the nonlinear effects associated with them should occur sooner than others. The importance of resonance triads for the explanation of the development of three-dimensional structures in a layer and the generation of intense pulsations has already been emphasized in [5, 6]. The clarification of the properties of the evolution of resonantly interacting disturbances therefore is important for an understanding of this transitional process. © 1979 Plenum Publishing Corporation.